We need to show that \(\Lambda _1^T(i)\)  does not depend on \(T\) . For any \(T_1> T\) , there exists a process \(\Lambda _1^{T_1}(i)\in L^2_{\mathcal {F}^W}(0,T_1;\mathbb {R}^n)\)  such that \((P_1^{T_1}(i),\Lambda _1^{T_1}(i))\)  is a solution to the \(i^{\mbox{th}}\)  equation in ESRE (REF ) on the time interval \([0,T_1]\)  with terminal value \(P_1^{T_1}(T_1,i)\) . Notice that \(P_1^{T_1}(T,i)=\lim \limits _{N\rightarrow \infty }P_{1,N}(T,i)=P_1^{T}(T,i)\) , by the uniqueness of the solution of the \(i^{\mbox{th}}\)  equation in ESRE (REF ) with terminal value \(P_1^{T_1}(T,i)=P_1^{T}(T,i)\)  (see Theorem 3.5 of [1]}), we get \(\Lambda _1^{T}(t,i)=\Lambda _1^{T_1}(t,i), \ t\in [0,T]\) .
